Positive definite and related functions on semigroups

Berg, Christian

doi:10.1515/9783110856040.253

Cited by 9 publications

(5 citation statements)

References 38 publications

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“…The SαS laws in this case have the first coordinate which is stable in the conventional sense in R, and the max-stable second coordinate. Thus, the parameter of such a stable law belongs to (0,2].…”

Section: Cones Violating Basic Assumptionsmentioning

confidence: 99%

“…However, not all semigroups possess a separating family of characters. For instance, if x + x = y + y and x + x + x = y + y + y for some x = y, then x and y cannot be separated by any character, since every character χ necessarily satisfies χ(x) 2 = χ(y) 2 and χ(x) 3 = χ(y) 3 , see Example 8.19.…”

Section: Characters On Semigroupsmentioning

confidence: 99%

“…for each Lévy measure λ. It is shown in [6] that a Lévy function exists, can be chosen to be continuous with respect to its second argument and to satisfy L(χ, ρ) = −L(χ, ρ), see also [2,Th. 3.1].…”

Section: Integral Representations Of Negative Definite Functionsmentioning

confidence: 99%

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Strictly stable distributions on convex cones

Davydov

Molchanov

Zuyev

2008

Electron. J. Probab.

129

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Using the LePage representation, a symmetric α-stable random element in Banach space B with α ∈ (0, 2) can be represented as a sum of points of a Poisson process in B. This point process is union-stable, i. e. the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes.These concepts makes sense in any convex cone, i. e. in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. We prove that random samples (or binomial point processes) in rather general cones converge in distribution in the vague topology to the union-stable Poisson point process. This convergence holds also in a stronger topology, which implies that the sums of points converge in distribution to the sum of points of the union-stable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with α-stable limit for α ∈ (0, 1).By using the technique of harmonic analysis on semigroups we characterise distributions of α-stable random elements and show how possible values of the characteristic exponent α relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. It is shown that several conditions imply that a stable random element admits the LePage representation. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.

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Section: Cones Violating Basic Assumptionsmentioning

confidence: 99%

Section: Characters On Semigroupsmentioning

confidence: 99%

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Strictly stable distributions on convex cones

Davydov

Molchanov

Zuyev

2008

Electron. J. Probab.

129

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“…Bisgaard and Ressel proposed to extend the concept of perfect * -semigroup (see [5]). In [1], this new terminology was introduced as perfect, and our terminology "perfect" was called Radon perfect. Bisgaard proved that every Radon perfect * -semigroup is perfect (see [4,Corollary 4.1]).…”

Section: ρ(S) Dµ(ρ) For S ∈ S mentioning

confidence: 99%

Perfectness of Conelike *-Semigroups in ℚk

Nishio

Sakakibara

2000

Math. Nachr.

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“…However, a comprehensive theory of integral representations of positive definite functions on semigroups exists. For this subject, see Berg, Christensen, and Ressel [2] and Berg [1].…”

Section: Introductionmentioning

confidence: 99%